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Einstein manifolds. Reprint of the 1987 edition. (English) Zbl 1147.53001
Classics in Mathematics. Berlin: Springer (ISBN 978-3-540-74120-6/pbk). xii, 516 p. (2008).
The monograph under review is a reprint of the 1987 Edition [Berlin etc.: Springer (1987; Zbl 0613.53001)].
The author decided to write a monograph on Einstein manifold in September 1979, at a symposium on Einstein manifolds held at Espalion, France, because most basic questions were still open, but good progress had been made, due in particular to the solution of Calabi’s conjecture by T. Aubin and S. T. Yau and to N. Koiso’s results on the moduli space.
Einstein manifolds are not only interesting in themselves, but are also related to many important topics of Riemannian geometry, for example, Riemannian submersions, homogeneous Riemannian spaces, Riemannian functionals and their critical points, Yang-Mills theory, holonomy groups, algebraic geometry via \(K3\) surfaces.
The present book is intended to be a complete reference book. It is structured into 17 chapters, an appendix on Sobolev spaces and elliptic operators and an addendum of results on five topics, too recent to include them into the main text. The fact that four of these topics are related to Kähler-Einstein manifolds is indicative of the permanently-developing links between complex and Riemannian geometry. The various chapters are self-contained.
The introduction starts with brief definitions and motivation for the study of Einstein manifolds. Interesting standard examples of Einstein manifolds are given.
Let \((M,g)\) be a Riemannian manifold. The curvature of \(g\) determines a symmetric bilinear form on the tangent space \(T_pM\), \(p\in M\), namely, the Ricci tensor \(r\). The metric \(g\) is said to be Einstein when \(r\) is a constant multiple of \(g\).
Chapters 1–3 deal with basic material on Riemannian and pseudo-Riemannian manifolds, Kähler manifolds and relativity.
Riemannian functionals are studied in Chapter 4. Since the work of Lagrange it is well-known that the equations of classical mechanics can be obtained as solutions of a variational problem, using a suitable functional. Then, a natural way to prove the existence of Einstein metrics appears. Einstein metrics are the solutions to the Euler-Lagrange equations for stationary points of the integral of the scalar curvature \(s\).
In Chapter 5, the Ricci curvature \(r\) of a metric \(g\) is considered as a partial differential operator. Conversely, given a Ricci candidate \(r\), is there a metric \(g\) whose Ricci curvature is exactly \(r\)? What conditions on \(r\) insure the uniqueness of such a metric? In particular, the above problem is studied for Einstein metrics.
The topology of an Einstein manifold is investigated in Chapter 6.
More explicit classes of Einstein metrics arise on homogeneous spaces \(G/H\). If \(G\) is a compact Lie group, any orbit of the adjoint representation admits a canonical Kähler-Einstein metric with \(s>0\). On the other hand, if \(H\) has irreducible isotropy representation, an invariant Riemannian metric \(g\) on \(G/H\) is forced to be proportional to its Ricci tensor \(r\). Further progress towards a classification of homogeneous Einstein manifolds has been made, particularly by M. Wang and W. Ziller.
It is known that the complex projective space \(\mathbb CP^3\) has a nonstandard homogeneous Einstein metric, a fact that may be explained in terms of a Riemannian submersion \(\pi:\mathbb CP^3\to S^4\). Such techniques were used by L. Bérard-Bergery to generalize an example due to D. Page of an Einstein metric on the connected sum \(\mathbb CP^2\,\#\,\overline{\mathbb CP^2}\).
The description of the above examples and methods forms the central part of the book (chapters 7, 8, 9 and 11). With the exception of a few difficult proofs, the exposition is self-contained, and contains a series of previously unpublished results.
Chapter 12 covers relevant, but more specialized topics. A sequel to the variational approach is provided by a study of deformations of Einstein metrics and work of N. Koiso on the moduli space problem. Chapters 10, 13 and 14 deal with holonomy groups, self-duality in four dimensions and quaternionic manifolds, respectively. Chapter 15 contains a rather brief report on important constructions in the noncompact case.
Finally, there is a discussion of generalizations of the Einstein condition; interesting though these are, the reader may be left in agreement with the author’s thesis that Einstein metrics are the nicest sort.
The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title.

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C30 Differential geometry of homogeneous manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C20 Global Riemannian geometry, including pinching